Are specific characteristics required for the person who places the deceased in the grave, and what should they do when burying the deceased, given that the person who places the deceased in the grave is the gravedigger and does not pray?

The people with the most right to bury the deceased and place them in their grave are the closest mahārim (unmarriageable kin), starting with the father, then the grandfather, then the son, and so on. The husband is given precedence over the mahārim in burying his wife according to Malik and Shafi'i, while the mahārim are given precedence over the husband according to Abu Hanifa and one narration from Ahmad.

As for the manner of burial, the grave is dug and a niche (laḥd) is made. The deceased is entered from the direction of the qiblah, and upon placing them, one says: "In the name of Allah and upon the creed of the Messenger of Allah." They are then laid on their right side with their face towards the qiblah. The knots (of the shroud) are untied, and the bricks are arranged over the laḥd.

One who does not pray has no right to place the deceased in their grave.